The third step aims to compress the information provided by the snapshots through either linear (yellow) or nonlinear (orange) compression. The linear compression employs the proper orthogonal decomposition (POD) technique [1]}, [2]}, [3]}. POD is used to determine characteristic spatial modes based on relative energy content criteria [4]}, [5]}, [6]}. During the linear compression, only the first \(\mathrm {N}\)  modes are retained [7]}, and employed as basis functions for the reduced basis spaces \(\mathcal {U}_\mathrm {N}\) , \(\mathcal {P}_\mathrm {N}\) , \(\mathcal {T}_\mathrm {N}\) , used for approximating the velocity, pressure, and temperature fields respectively. The goal is to achieve \(\mathrm {N} \ll \mathrm {M} N^t\)  as well as \(\mathrm {N} \ll N_h^u\) , \(\mathrm {N} \ll N_h^p\) , and \(\mathrm {N} \ll N_h^T\) . The resulting spatial modes, denoted by \(\mathbf {w}_1^{\mathbf {u}}, \cdots , \mathbf {w}_\mathrm {N}^{\mathbf {u}}\) , \(\mathbf {w}_1^p, \cdots , \mathbf {w}_\mathrm {N}^p\) , and \(\mathbf {w}_1^T, \cdots , \mathbf {w}_\mathrm {N}^T\)  span a linear subspace of the velocity, pressure, and temperature fields respectively.
